Ann. Inst. H. Poincaré Probab. Statist. 52, no. 3, 1406-1436
We are interested in random loops that arise as the scaling limits of statistical mechanics interfaces. Typically, one cuts up a domain of the plane by a lattice of very small mesh size, and randomly colors in black and white the lattice faces according to a certain law. The objects of interest are interfaces i.e. path that separate black and white regions. Such loops are expected to be conformally invariant in the scaling limit. Moreover, one may be able to explicitly compute how these random loops vary when the boundary of the domain is perturbed. Conjecturally, any such loop measure fall in a one parameter family indexed by the central charge. In particular, conjecturally, random curves are characterized by their covariance property, i.e. by their interaction with the boundary.
Werner showed that there is a unique way to associate to any Riemann surface a measure on its simple loops, such that the collection of measures satisfy a strong conformal invariance property (no interaction with the boundary). These random loops are constructed as the boundary of Brownian loops, and correspond in the zoo of statistical mechanics models to central charge $0$, or Schramm-Loewner Evolution (SLE) parameter $\kappa=8/3$.
The goal of this paper is to construct a family of measures on simple loops on Riemann surfaces that satisfies a conformal covariance property, and that would correspond to SLE parameter $\kappa=2$ (central charge $-2$).